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Mass localization

For a given class $\mathcal{F}$ of closed sets of a measured metric space $(E,d,μ)$, we want to find the smallest element $B$ of the class $\mathcal{F}$ such that $μ(B)\geq 1-α$, for a given $0<α<1$. This set $B$ \textit{localizes the mass} of $μ$. Replacing the measure $μ$ by the empirical measure $μ_n$ gives an empirical smallest set $B_n$. The article introduces a formal definition of small sets (and their size) and study the convergence of the sets $B_n$ to $B$ and of their size.